Cyclic spaces and S^1-equivariant homotopy theory

Question

I'm trying to understand the relationship between cyclic spaces and S¹-equivariant homotopy theory. More precisely, I only care about S¹-spaces up to equivalence of fixed point spaces for the finite subgroups of S¹. Given a cyclic space X : ΔC^op → Top, I know the geometric realization of the restriction of X to Δ^op is an S¹-space. Form the associated fixed point diagram O^op → Spaces where O is the full subcategory of the orbit category of S¹ on the objects S¹/C where C ranges over finite subgroups of S¹. I regard the category of functors O^op → Spaces as an (∞,1)-category.

My question is, what structure on X does the resulting diagram depend on? More specifically, under what conditions does a map f : X → Y of cyclic spaces induce an equivalence of fixed point diagrams?

In O consider the full subcategory O₁ on the object S¹/{•}. The restriction of this diagram to O₁ is a space with S¹-action in the (∞,1)-categorical sense, and I think it's just the left Kan extension of X along the functor ΔC^op → BS¹ induced by the fact that ΔC is the quotient of something (ΔZ) by an S¹-action. Thus it only depends on X viewed as a functor from ΔC^op to the (∞,1)-category of spaces. But to evaluate on the other objects of O, corresponding to the fixed point spaces of nontrivial finite subgroups of S¹, do I need to know each X[r] as a C_r+1 space (i.e. the homotopy types of the fixed points sets for subgroups of C_r+1)? Is there a way to encode all of that information in a functor from some (maybe (∞,1)-)category to Spaces? Or is it possible that I need to remember even more information about X?

Edit: I guess another way to phrase the question is this: I'm looking for a model category structure on the category of functors ΔC^op → Top, such that the identity functor to the injective model structure is a left Quillen functor, and such that the geometric realization to genuine S¹-spaces is also a left Quillen functor. Furthermore I would like to know whether this model category structure is Quillen equivalent to a diagram category of spaces (possibly on a topological index category) with objectwise weak equivalences.

Answer 1

I don't know if this is exactly what you're looking for (and there's a good chance you already know what I'm going to write) but let me give it a try:

The realization functor of cyclic sets (not spaces!) to $S^1$-spaces can be made part of a Quillen equivalence for two of the three commonly desired model structures on $S^1$-spaces: The model structure that gives you "Spaces over $BS^1$" is given in a 1985 paper of Dwyer-Hopkins-Kan, while a model structure that gives you the equivalences that you want (i.e., checked on fixed sets for finite subgroups) is given in a 1995 paper "Strong homotopy theory of cyclic sets" by Jan Spalinksi.

(Irrelevant to your question, but along the same lines: A recent paper of Andrew Blumberg describes how one can throw in some extra--still combinatorial--data and obtain a combinatorial model of the third desirable model structure on $S^1$-spaces, namely where equivalences are those that induce equivalences on fixed sets for all closed subgroups.)

Spalinksi's model structure depends on the following construction of $|X_{.}|^{C_n}$ (as a space-over-$BS^1$) in terms of the subdivision construction: The simplicial set $(sd_r X)_n = X_{r(n+1)-1}$ has an action of $C_r$--since $C_r$ is a subgroup of the copy of $C_{r(n+1)}$ acting on $X_{r(n+1)-1}$; taking fixed points (in sSet) and then realizing gives $|X_{.}|^{C_n}$.

This suggests (though I haven't checked too carefully) that remembering each $X_n$ as a $C_{n+1}$-space (in the sense you suggest, with subgroups) is enough, as you expected.

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Now begins the speculative (and probably wrong) part of this answer: I have nothing too certain to say about writing this as a functor category, but it doesn't seem too unreasonable (to me, right now, at least) based on the above simplicial subdivision construction that we might be able to construct a reasonable candidate: some sort of mix of the cyclic category and the orbit categories for the cyclic groups. Purely combinatorially, this seems to get tricky.

But, I think we can realize this geometrically: Let $(S^1)_r$ be the circle equipped with a $\mathbb{Z}$ action given by the rotation by $2\pi/r$. We could try to define $Hom'([m-1]_r, [m'-1]_{r'})$ along the lines of "(htpy classes of) degree $r'/r$, increasing $\mathbb{Z}$-equivariant maps $S^1 \to S^1$ sending the $mr$-torsion points to the $m' r'$-torsion points". This should correspond to taking all the $r$-cyclic categories and sticking them together, and in particular is bigger than what we want. But, the $\mathbb{Z}$-action on the circles should induce one on the $Hom'$-sets and the composition should respect it. Taking the quotient, we seem to get something that looks like a reasonable candidate. For each fixed $r$, we should be getting a copy of the cyclic category. And, e.g. $Hom([m-1]_r, [mr-1]_1)$ should contain $Hom_{orbit}(Z/mr, Z/r)$. (Disclaimer: It's late and I haven't checked any of this too carefully!)